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Katie’s Binary Nails Tutorial – and a puzzle

I’ve just posted my latest YouTube video, in which I explain how to use binary numbers to jazz up your nail varnish:

Alongside this video, I also have an associated puzzle for you to think about.

At the very end of the video, there’s a clip showing my hand with every possible combination of varnish from 0 to 31 – shown in this gif:

The puzzle is:

Suppose I want to paint my nails on one hand differently every day for a month – so I need to use all 31 combinations involving glitter. Assuming that a nail painted with plain varnish can have glitter added, but a nail with glitter needs to be nail-varnish-removed before it can become a plain nail again, what order do I apply the different combinations so that you minimise the amount of nail varnish remover I’ll need to use?

For example, you could paint on 1 (00001) for the first day, then quite easily change it to a 3 (00011) for day 2 by glittering one nail, but to do 2 (00010) would require removing glitter. In taking the photos to make the video clip, I came up with a quick ordering on the back of an envelope, which did save me some time and effort from doing it in number order, but I’m pretty sure it’s not optimal. Can you find a better one?

We’ll be posting the answer, with the interesting maths behind all this, in the next week or so. In the meantime, no spoilers in the comments (here or on the video!)


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